Newform invariants
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below.
We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} - 11 x^{6} + 9 x^{5} + 34 x^{4} - 19 x^{3} - 27 x^{2} + 11 x - 1\):
\(\beta_{0}\) | \(=\) | \( 1 \) |
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{3} - 5 \nu \) |
\(\beta_{3}\) | \(=\) | \((\)\( -2 \nu^{7} + \nu^{6} + 23 \nu^{5} - 8 \nu^{4} - 76 \nu^{3} + 12 \nu^{2} + 65 \nu - 7 \)\()/2\) |
\(\beta_{4}\) | \(=\) | \( -\nu^{7} + \nu^{6} + 11 \nu^{5} - 9 \nu^{4} - 34 \nu^{3} + 18 \nu^{2} + 27 \nu - 7 \) |
\(\beta_{5}\) | \(=\) | \((\)\( -4 \nu^{7} + 3 \nu^{6} + 45 \nu^{5} - 26 \nu^{4} - 144 \nu^{3} + 50 \nu^{2} + 121 \nu - 27 \)\()/2\) |
\(\beta_{6}\) | \(=\) | \((\)\( 5 \nu^{7} - 4 \nu^{6} - 55 \nu^{5} + 34 \nu^{4} + 170 \nu^{3} - 61 \nu^{2} - 136 \nu + 27 \)\()/2\) |
\(\beta_{7}\) | \(=\) | \((\)\( -6 \nu^{7} + 5 \nu^{6} + 67 \nu^{5} - 42 \nu^{4} - 212 \nu^{3} + 72 \nu^{2} + 175 \nu - 29 \)\()/2\) |
\(1\) | \(=\) | \(\beta_0\) |
\(\nu\) | \(=\) | \(\beta_{1}\) |
\(\nu^{2}\) | \(=\) | \(\beta_{5} - \beta_{4} - \beta_{3} - \beta_{1} + 3\) |
\(\nu^{3}\) | \(=\) | \(\beta_{2} + 5 \beta_{1}\) |
\(\nu^{4}\) | \(=\) | \(\beta_{7} + 6 \beta_{5} - 8 \beta_{4} - 7 \beta_{3} - 7 \beta_{1} + 15\) |
\(\nu^{5}\) | \(=\) | \(\beta_{7} + 2 \beta_{6} + \beta_{5} + 8 \beta_{2} + 28 \beta_{1} + 1\) |
\(\nu^{6}\) | \(=\) | \(11 \beta_{7} + 2 \beta_{6} + 37 \beta_{5} - 54 \beta_{4} - 48 \beta_{3} - 47 \beta_{1} + 86\) |
\(\nu^{7}\) | \(=\) | \(13 \beta_{7} + 24 \beta_{6} + 12 \beta_{5} - \beta_{4} - 3 \beta_{3} + 54 \beta_{2} + 163 \beta_{1} + 9\) |
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
This newform does not admit any (nontrivial) inner twists.
\( p \) |
Sign
|
\(3\) |
\(-1\) |
\(5\) |
\(-1\) |
\(89\) |
\(1\) |
This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4005))\):